Current Loops and fundamental group

[wofsy]

The Law of Biot and Savart Law tells us how to find a differential form that generates the first de Rham cohomology of S3- embedded loop. Run a steady current through the loop.

This form is just the dual of the induced magnetic field (using the Euclidean metric).

Ampere's Law tells us that the form is closed. In fact, integration of the form over small cycles that lie in a tube around the circuit and are orthogonal to the current is 1 - so the form is a generator of the first cohomology.

What physical phenomenon tells us the fundamental group of S3 - embedded loop?

 

[fantispug]

That's a clever question, but I'm not sure if there is an answer. The electric field (by itself) only tells us about the second de Rham cohomology, and the gravitational field would do the same.

From the classical physics I remember only the de Rham cohomolgy came up (when using Stokes' theorem and trying to construct potentials); I can't think of an interpretation of the fundamental group. You might have more luck in quantum physics or string theory.

Sorry I can't be of more help.

Can you extend it another way? Can you find analogues of the electric and magnetic fields in higher dimensions to calculate de Rham cohomologies of S^n - embedded submanifold?

 

[wofsy]

That's a clever question, but I'm not sure if there is an answer. The electric field (by itself) only tells us about the second de Rham cohomology, and the gravitational field would do the same.

From the classical physics I remember only the de Rham cohomolgy came up (when using Stokes' theorem and trying to construct potentials); I can't think of an interpretation of the fundamental group. You might have more luck in quantum physics or string theory.

Sorry I can't be of more help.

Can you extend it another way? Can you find analogues of the electric and magnetic fields in higher dimensions to calculate de Rham cohomologies of S^n - embedded submanifold?

Thanks for the reply and the problem. I haven't totally worked out the higher dimensional generalization of a magnetic field induced by a q dimensional current yet but felt like not ignoring your post. So here's an idea.

M is a smoothly embedded q dimensional oriented submanifold w/o boundary in Euclidean n-space.

For any point x not on M and any point m on M, the vector x-m and the volume element,w, of M determine an n-q-1-form at m that is perpendicular to the q+1 plane that (x-m) and w determine and has (q-1) volume equal to the length of x-m. Choose the form so that (x-m),w, and the form taken together give the positive orientation of R^n at m.

This form seems like a reasonable analogue of the cross product.

The form needs to be scaled before integration. After scaling, integrate this n-q-1 form over M to get an analogue of a magnetic field induced by a steady q dimensional "current" on M.

If the scaling is analgous to that used in the Law of Biot and Savart then for points very close to M the field should look like the volume element of the n-q-1 sphere that is perpendicular to the nearest tangent plane of M. Haven't yet worked out the formula.

Thanks again for this. I am learning a lot.

By the way, yesterday I sat in on a lecture on quantum computers. The speaker was a mathematician who is part of a research team at Microsoft. He spoke about things I don't even begin to understand but it was all about Topological Quantum Field Theories and involved braids and links in 3 space. He spoke of Jones polynomials, braid groups, and Chern-Simons theories. So that stuff may be a physical counterpart to the fundamental group of the complement of a circuit or of links. Do you know about this stuff?